Question: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$4.50$ each for students, and the group paid $$51.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$20.00$ each for teachers and $$9.50$ each for students, and the group paid $$136.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+4.5y = 51}$ ${20x+9.5y = 136}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-18y = -204}$ ${20x+9.5y = 136}$ Add the top and bottom equations together. $ -8.5y = -68 $ $ y = \dfrac{-68}{-8.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5x+4.5y = 51}$ to find $x$ ${5x + 4.5}{(8)}{= 51}$ $5x+36 = 51$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {20x+9.5y = 136}$ and get the same answer for $x$ ${20x + 9.5}{(8)}{= 136}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.